Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #272923 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,12,9,11)(8,10)(13,14)(15,22)(16,24)(17,26)(18,20)(19,23)(21,27)(25,28);(1,2)(3,4)(5,7,10,9)(6,11,8,12)(13,15,18,16)(14,19,20,28)(17,25,27,23)(21,24,26,22);(2,4)(5,8,10,6)(7,12,9,11)(14,20)(19,28)(21,26)(22,24);(5,9,10,7)(6,12,8,11)(13,16,18,15)(14,19,20,28)(17,23,27,25)(21,24,26,22)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=hi,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 180180);